What is the right way to think of omitted variable bias in a regression that only has dummy variables?
Let's say I have the following equation:
where y is the price of shoes; and x1, x2 and x3 are dummy variables for three different regions (x4, the reference region was omitted).
And I suspect that I am missing a variable (x4) to adjust for 'level of urbanization of each region' -- each region contains an uneven number of areas with different levels of urbanization. Thus, the true model is, or so I suspect:
Now I know I can sign the bias of any one of the coefficients in equation (1) if two conditions are met: a) x4 is correlated with either x1, x2 or x3; and b) x4 has an impact on y (i.e. δ>0).
However, I am not sure if I can explore condition a) above since the correlation of x4, in this case, would be with a categorical variable, not a continuous one.
How can I go about this?